9 METRIZABLE GROUPS 43
distance which is both left and right-invariant (cf. Section 14.3, Problem 11).) (Use
the fact that if the topology of a metrizable group G can be defined by a distance
which is both left and right-invariant, then for every neighborhood V of the neutral
element e of G there exists a neighborhood W <= V of e such that xWx'1 c V for all
2. Let S be a compact subset of a metrizable group G, such that xy e S whenever x e S
and y e S. Show that for each x e S we have xS = S. (Consider a cluster value y of the
sequence (xn}n^i in S, and show that yS is the intersection of the sets x"S; deduce that
yxS = yS.) Deduce that S is a subgroup of G.
3 Let G be a locally compact and totally disconnected metrizable group.
(a) Show that every neighborhood of e in G contains an open compact subgroup
of G. (Every neighborhood V of e contains a neighborhood U of e which is both open
and closed (Section 3.19, Problem 9). If B = ()U, show that there exists a symmetric
open neighborhood W of e in G such that W <=• U and UW n BW = 0, and deduce
that the subgroup generated by W is contained in U.)
(b) Suppose that the topology of G can be defined by a left- and right-invariant
distance. Show that every neighborhood of e in G contains a compact open normal
subgroup of G (remark that every neighborhood V of e contains a symmetric open
neighborhood W such that xWx~l <= V for all x e G).
4. Let p be a prime number. Consider the family of finite groups Z/p"Z (« ^ 1), each
with the discrete topology, and their product G (Section 12.8, Problem 8) which is
compact and totally disconnected. For each «, let ^pn:Z/paZ^ZIpn~"iZ be the
canonical homomorphism.
(a) Show that the set of all z — (zn) E G such that <pa(zn) — zn_i for all n is a closed
(hence compact) subgroup Zp of G. If </rn is the restriction to Zp of the projection
prn : G ->• Z/p"Z, then i/ftt is a surjective homomorphism.
(b) For each z = (zn) e Zp , put \z\p = 0 if z = 0, and |z|p = pl ~m if m is the smallest
integer such that zm^0. Show that \z—z'\p is a translation-invariant distance
which defines the topology of Zp .
(c) For each n^>lt let fn : Z-*Z/p"Z be the canonical homomorphism. Show that
the homomorphism/: zt—^(fa(z))n^i of Z into Zp is injective and that its image /(Z)
is a dense subgroup of Zp. If we put d(z, z') = \f(z) — /(z')|P, show that d is the
^p-adic distance on Z (3.2.6). The elements of Zp are called />-adic integers.
5. Let p be a prime number, and let G be the compact group which is the product of an
infinite sequence (Gn)n>o of groups Gn all equal to T =R/Z. For each n, let <pn be the
homomorphism x\-+px of Gn into Gn-i- The compact subgroup of G consisting of all
z = (zn) such that <pn(zn) = zn-i for all n is called the p-adic solenoid and is denoted
byTp.
(a) For each n9 let fn : Tp -> Gtt = T be the restriction of prn to Tp . Show that fn is a
surjective homomorphism with kernel isomorphic to the group Zp (Problem 4).
(b) Let 9? : R -» T be the canonical homomorphism. For each x e R, show that the
point B(x)=(cjp(xlprf))n^Q belongs to Tp. Prove that 6 is an injective continuous
homomorphism of R into Tp, and that ^(R) is a dense subgroup of Tp . Deduce that
Tp is connected.
(c) Let I be an open interval in R, with centre 0 and length <1. Show that the
subspace fo1(<p(T)) of Tp is homeomorphic to the product I x Zp. Deduce that Tp
is not locally connected.n of {x} and the set of integers J>/i,